Time-Space Tradeoffs for Learning from Small Test Spaces: Learning Low Degree Polynomial Functions

We develop an extension of recently developed methods for obtaining time-space tradeoff lower bounds for problems of learning from random test samples to handle the situation where the space of tests is signficantly smaller than the space of inputs, a class of learning problems that is not handled by prior work. This extension is based on a measure of how matrices amplify the 2-norms of probability distributions that is more refined than the 2-norms of these matrices. As applications that follow from our new technique, we show that any algorithm that learns $m$-variate homogeneous polynomial functions of degree at most $d$ over $\mathbb{F}_2$ from evaluations on randomly chosen inputs either requires space $\Omega(mn)$ or $2^{\Omega(m)}$ time where $n=m^{\Theta(d)}$ is the dimension of the space of such functions. These bounds are asymptotically optimal since they match the tradeoffs achieved by natural learning algorithms for the problems.